manifolds and cobordisms
cobordism theory, Introduction
A hyperbolic space is the analog of a Euclidean space as one passes from Euclidean geometry to hyperbolic geometry. The generalization of the concept of hyperbolic plane to higher dimension.
A hyperbolic manifold is a Riemannian manifold $(X,g)$ of constant sectional curvature $-1$.
There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.
There is a curious relation of volumes of hyperbolic 3-manifolds to the action functional of Chern-Simons theory/Dijkgraaf-Witten theory.
Let $G$ be a Lie group and $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ a cocycle in degree-3 (generalized) Lie group cohomology. Write $\flat G$ for the underlying discrete group and $\flat \mathbf{c} \colon \mathbf{B} \flat G \to \mathbf{B}^3 \flat U(1)$ for the induced cocycle in ordinary (discrete) group cohomology, $[\flat \mathbf{c}] \in H^3_{Grp}(G_{disc},U(1)_{disc})$.
Then for $\Sigma$ a closed manifold of dimension 3, a map (of smooth infinity-groupoids) $\Sigma \to \mathbf{B}\flat G$ is a flat $G$-principal connection on $\Sigma$ and the composite
is the action functional for $G$-Chern-Simons theory on $\Sigma$ restricted to $G$-flat connections, or equivalently is the action functional of $\flat G$-Dijkgraaf-Witten theory.
Now for $G = SL(n,\mathbb{C})$ the complex special linear group and hence for Chern-Simons theory with complex gauge group, it turns out that the imaginary part of this flat Chern-Simons/Dijkgraaf-Witten invariant of 3-manifolds always has an expression as a combination of volumes of hyperbolic 3-manifolds.
For $n = 2$ this is well understood conceptually. For $n \geq 3$ it has been checked by the computer algebra? system SnapPy (Zickert 07), but the conceptual reason remains unclear.
The combination $CS(\omega) + i vol$ is also called the complex volume or Cheeger-Simons class of a hyperbolic 3-manifold (e.g. Neumann 11, section 2.3. Garoufalidis-Thurston-Zickert 11).
(It is maybe noteworthy that the same kind of combination appears as the contribution of membrane instantons.)
The Mostow rigidity theorem states that every hyperbolic manifold of dimension $\geq 3$ and of finite volume is uniquely determined by its fundamental group.
A Riemannian manifold
with zero sectional curvature is a Euclidean manifold?;
with +1 sectional curvature is an elliptic manifold?
See also
See also
Wikipedia, Hyperbolic manifold
Wikipedia, Hyperbolic space
The relation of volumes of hyperbolic 3-manifolds to analytically continued Chern-Simons theory/Dijkgraaf-Witten theory is discussed in
Walter Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413-474 (arXiv:math/0307092)
Christian Zickert, The volume and Chern-Simons invariant of a representation, Duke Math. J., 150 (3):489-532, 2009 (arXiv:0710.2049)
Walter Neumann, Realizing arithmetic invariants of hyperbolic 3-manifolds, Contemporary Math 541 (Amer. Math. Soc. 2011), 233–246 (arXiv:1108.0062)
Stavros Garoufalidis, Dylan Thurston, Christian Zickert, The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds (arXiv:1111.2828, Euclid)
Last revised on April 24, 2018 at 10:38:57. See the history of this page for a list of all contributions to it.